Type any trigonometry problem into the solver above and get a clear, step-by-step solution in seconds. From right triangles to trig equations and identities, you will see exactly how each answer is reached.
Free · Step-by-step worked solutions · Works on any device
What this trigonometry solver does
This free trigonometry solver works through trig problems one step at a time, so you do not just get the final number — you see the reasoning behind it. Enter your problem as text or build it with the equation editor, and our AI engine returns a clean, organized solution.
It handles the topics you meet from geometry through pre-calculus, including:
- Right triangles: finding missing sides and angles with SOH CAH TOA.
- Evaluating functions: exact values like \( \sin 30^\circ \) or \( \cos\frac{\pi}{4} \), plus decimal approximations.
- Trigonometric equations: solving for an angle, such as \( 2\sin x - 1 = 0 \).
- Identities: using the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \) and others to simplify expressions.
- Laws of Sines and Cosines for non-right triangles.
Worked examples
Example 1: Find a missing side in a right triangle
A right triangle has an angle of \( 38^\circ \) and a hypotenuse of \( 9 \) cm. Find the side opposite that angle.
The opposite side and the hypotenuse are linked by sine, so use \( \sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}} \):
$$\sin 38^\circ = \frac{x}{9}$$Multiply both sides by \( 9 \):
$$x = 9 \sin 38^\circ \approx 9 \times 0.6157 \approx 5.54$$Example 2: Solve a trigonometric equation
Solve \( 2\sin x - 1 = 0 \) for \( 0^\circ \le x < 360^\circ \).
First isolate \( \sin x \):
$$2\sin x = 1 \quad\Rightarrow\quad \sin x = \frac{1}{2}$$The reference angle is \( 30^\circ \). Since sine is positive in the first and second quadrants, there are two solutions:
$$x = 30^\circ \qquad x = 180^\circ - 30^\circ = 150^\circ$$Common mistake
Do not stop at the first answer. Sine, cosine, and tangent repeat, so most trig equations have more than one solution within a full \( 360^\circ \) rotation.
Example 3: Use the Pythagorean identity
Given \( \sin\theta = \dfrac{3}{5} \) with \( \theta \) in the first quadrant, find \( \cos\theta \) and \( \tan\theta \).
Start from \( \sin^2\theta + \cos^2\theta = 1 \) and solve for \( \cos\theta \):
$$\cos\theta = \sqrt{1 - \left(\tfrac{3}{5}\right)^2} = \sqrt{1 - \tfrac{9}{25}} = \sqrt{\tfrac{16}{25}} = \frac{4}{5}$$We take the positive root because \( \theta \) is in the first quadrant. Then:
$$\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{3/5}{4/5} = \frac{3}{4}$$How to use the trigonometry solver
- Enter your problem. Type it in plain text or use the equation editor to add angle symbols, fractions, and trig functions exactly as they appear in your homework.
- Set degrees or radians. Make sure the angle mode matches your problem so values like \( \sin 90 \) are interpreted correctly.
- Read the steps. Review each line of the worked solution, then check the highlighted final answer and try a similar problem to lock in the method.
Key formula reminder
SOH CAH TOA tells you which ratio to use: \( \sin = \dfrac{\text{opp}}{\text{hyp}} \), \( \cos = \dfrac{\text{adj}}{\text{hyp}} \), \( \tan = \dfrac{\text{opp}}{\text{adj}} \).
Related step-by-step guides
- SOH CAH TOA: How to Solve Trig Problems — a beginner-friendly walkthrough of the three core ratios.
- How to Solve Trigonometric Equations — find every solution in a given interval with confidence.
Keep solving
Trig problems often sit inside larger questions. When you need broader support, try our Equation Solver or work through related topics with the Algebra Solver.
